#265022

order	512
nilpotency class	3
isoclinism	isoclinic class of #265019
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1059 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(2,6)(3,4)(7,8)(9,10,16,22)(11,19,25,34)(12,26,29,38)(13,30,24,17)(14,21,15,20)(18,32,36,28)(23,39,40,33)(27,37,31,35);(2,7)(3,4)(6,8)(9,11,14,24,16,25,15,13)(10,17,20,19,22,30,21,34)(12,23,28,27,29,40,32,31)(18,33,26,37,36,39,38,35);(1,2)(3,7)(4,6)(5,8)(9,12)(10,18)(11,23)(13,31)(14,28)(15,32)(16,29)(17,33)(19,37)(20,26)(21,38)(22,36)(24,27)(25,40)(30,39)(34,35);(2,8)(6,7)(9,13,15,25,16,24,14,11)(10,19,21,17,22,34,20,30)(12,27,32,23,29,31,28,40)(18,35,38,39,36,37,26,33)

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=eh,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,f^b=fhi,g^b=ghi,d^c=di,f^c=fi,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 172218);